Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 7.4, Exercise 7.4.7.
The postulates of quantum mechanics that we described earlier included specifications for the matrix elements of position and momentum in position space:
A more fundamental form of this postulate is to specify the commutation relation between and , which is independent of the basis and is
This allows the construction of explicit forms of the operators in other bases, such as the momentum basis, where
We can verify this by calculating the commutator by applying it to a function :
Thus 3 is satisfied in the momentum basis as well.
The standard recipe for converting a classical system to a quantum one is to first calculate the Poisson bracket for two physical quantities in the classical system, which gives
where and are the canonical coordinates and momenta. To convert to a quantum commutator, we replace the classical quantities by their quantum operator equivalents and the Poisson bracket by times the corresponding commutator. That is
For the case of and , we have, in classical mechanics in one dimension
so the quantum commutator is given by 3.
For other quantities, we can use the theorems on the Poisson brackets to reduce them:
Quantum commutators obey similar rules
The main difference between Poisson brackets and commutators is that, for the latter, the order of the operators in the last equation can make a difference. That is, in 14 we could also have written
since all three quantities are numerical (not operators), so multiplication commutes. In 17 it is not true in general that, for example
The conversion from classical to quantum mechanics can then be achieved in general by replacing
where each of the operators in the last equation is obtained by replacing in the first equation by and by . We do need to be careful with the ordering of the operators in the quantum version, however.
As an example, suppose we have
In the classical version, we calculate the Poisson bracket
Thus, by our rule above, the quantum version should be
We can verify this using 17
In this case, there is no ordering ambiguity in the quantum version, since is just a number.
For a second example, suppose we have
In the classical case, this result is the same as , but because and don’t commute in the quantum form, we need to be careful about the ordering.
We can do the calculation:
From 33 we have
so we get
Thus if the Poisson bracket involves a product of and , this should be replaced by
in the quantum version.